The Range of Variation of the Mass of the Most Massive Star in Stellar Clusters Derived from 35 Million Monte Carlo Simulations

A growing fraction of Simple Stellar Population (SSP) models, in an aim to create more realistic simulations capable of including stochastic variation in their outputs, begin their simulations with a distribution of discrete stars following a power-law function of masses. Careful attention is needed to create a correctly sampled Initial Mass Function (IMF) and in this contribution we provide a solid mathematical method called MASSCLEAN IMF Sampling for doing so. We then use our method to perform $10$ $million$ MASSCLEAN Monte Carlo stellar cluster simulations to determine the most massive star in a mass distribution as a function of the total mass of the cluster. We find a maximum mass range is predicted, not a single maximum mass. This maximum mass range is (a) dependent on the total mass of the cluster and (b) independent of an upper stellar mass limit, $M_{limit}$, for $unsaturated$ clusters and comes out naturally using our IMF sampling method. We then turn our analysis around, now starting with our new $25$ $million$ simulated cluster database, to constrain the highest mass star from the observed integrated colors of a sample of 40 low-mass LMC stellar clusters of known age and mass. Finally, we present an analytical description of the maximum mass range of the most massive star as a function of the cluster's total mass, and present a new $M_{max}-M_{cluster}$ relation.